Because the right hand side of (8) contains only first partial derivatives of u, it has the form of a Hamilton-Jacobi equation
of classical mechanics ,
This general relativistic Hamilton-Jacobi equation
becomes a scalar wave equation via the transformation to eliminate the squared first derivative, i.e., by defining the wave function [PSI] (q, p, t) of position q, momentum p, and time t as
Identifying [bar.p] with [nabla]S and comparing with the Hamilton-Jacobi equation
(b) Higher-order WKB corrections: Using WKB approximation method, the lowest order of the equation of motion describing a particle moving in the black hole gives the Hamilton-Jacobi equation
. The WKB approximation breaks down when the de Broglie wavelength of the particle, [[lambda].sub.p] ~ h/E, becomes comparable to the horizon of black hole, [r.sub.H].
By taking into account the effect of GUP, in a curved spacetime, the revised Hamilton-Jacobi equation
for the motion of scalar particles can be expressed as 
The main aim of this paper is to investigate thermal radiation from Klein-Gordon equation, Maxwell's electromagnetic field equations, Dirac particles, and also nonthermal radiation of Hamilton-Jacobi equation
in general nonstationary black hole.
At the classical limit (h [right arrow] 0) Q vanishes and (4) reduces to the Hamilton-Jacobi equation
. For this reason, Bohm  suggested that S is the classical action function, which relates to the actual velocity, v = [nabla]S /m, of the particle.
In this paper, we adopt a simple and effective method (i.e., Hamilton-Jacobi equation
) to analyze the Hawking radiation of the regular phantom BH.
The Hamilton-Jacobi equation
is derived for such motion and the effects of the curvature upon the quantization are analyzed, starting from a generalization of the Klein-Gordon equation in curved spaces.
Recall that the QCM general wave equation derived from the general relativistic Hamilton-Jacobi equation
is approximated by a Schrodinger-like wave equation and that a QCM quantization state is completely determined by the system's total baryonic mass M and its total angular momentum [H.sub.[summation]].
When the above expression for the Weyl scalar curvature (Bohm's quantum potential given in terms of the ensemble density) is inserted into the Hamilton-Jacobi equation
, in conjunction with the continuity equation, for a momentum given by [p.sub.k] = [[partial derivative].sub.k]S, one has then a set of two nonlinear coupled partial differential equations.
which is therefore but another form taken by the KG equation (as expected from the fact that the KG equation is the quantum equivalent of the Hamilton-Jacobi equation